Applications Of Derivatives Question 240
Question: Let $ h(x)=f(x)-{{(f(x))}^{2}}+{{(f(x))}^{3}} $ for every real number x. Then
[IIT 1998]
Options:
A) h is increasing whenever f is increasing
B) h is increasing whenever f is decreasing
C) h is decreasing whenever f is decreasing
D) Nothing can be said in general
Show Answer
Answer:
Correct Answer: A
Solution:
$ h(x)=f(x)-{{[f(x)]}^{2}}+{{[f(x)]}^{3}} $
$ h’(x)=f’(x)-2f(x)f’(x)+3{{[f(x)]}^{2}}f’(x) $
$ =f’(x)[1-2f(x)+3{{[f(x)]}^{2}}] $
$ =3f’(x){ {{( f(x)-\frac{1}{3} )}^{2}}+\frac{2}{9} } $
$ \therefore $ $ h’(x) $ and $ f’(x) $ have same sign.
BETA
